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1.7 Diagonal, Triangular, and Symmetric Matrices Diagonal Matrices (1/3) A square matrix in which all the entries off the main diagonal are zero is called a diagonal matrix . Here are some examples. A general n×n diagonal matrix D can be written as Diagonal Matrices (2/3) A diagonal matrix is invertible if and only if all of its diagonal entries are nonzero; Powers of diagonal matrices are easy to compute; we leave it for the reader to verify that if D is the diagonal matrix (1) and k is a positive integer, then: Diagonal Matrices (3/3) Matrix products that involve diagonal factors are especially easy to compute. For example, To multiply a matrix A on the left by a diagonal matrix D, one can multiply successive rows of A by the successive diagonal entries of D, and to multiply A on the right by D one can multiply successive columns of A by the successive diagonal entries of D . Example1 Inverses and Powers of Diagonal Matrices Triangular Matrices A square matrix in which all the entries above the main diagonal are zero is called lower triangular . A square matrix in which all the entries below the main diagonal are zero is called upper triangular . A matrix that is either upper triangular or low r triangular is called triangular . Example2 Upper and Lower Triangular Matrices Theorem 1.7.1 Example3 Upper Triangular Matrices Symmetric Matrices A square matrix A is called symmetric if A= AT. The entries on the main diagonal may b arbitrary, but “mirror images” of entries across the main diagonal must be equal. a matrix A aij is symmetric if and only if aij a ji for all values of i and j . Example4 Symmetric Matrices Theorem 1.7.2 AB T BT AT BA Recall : Since AB and BA are not usually equal, it follows that AB will not usually be symmetric. However, in the special case where AB=BA ,the product AB will be symmetric. If A and B are matrices such that AB=BA ,then we say that A and B commute . In summary: The product of two symmetric matrices is symmetric if and only if the matrices commute . Example5 Products of Symmetric Matrices The first of the following equations shows a product of symmetric matrices that is not symmetric, and the second shows a product of symmetric matrices that is symmetric. We conclude that the factors in the first equation do not commute, but those in the second equation do. Theorem 1.7.3 If A is an invertible symmetric matrix ,then A1 is symmetric. In general, a symmetric matrix need not be invertible. Products A A and AA T T The products AT A and AAT are both square matrices. ---- the matrix AAT has size m×m and the matrix AT A has size n×n . Such products are always symmetric since Example6 The Product of a Matrix and Its Transpose Is Symmetric Theorem 1.7.4 A is square matrix. If A is T AA an invertible matrix ,then and AT A are also invertible.